Question

The solution of `dy/dx = 1 + y + y^2 + x + xy + xy^2` is ______

विकल्प

• `2/sqrt3 tan^-1((2y + 1)/sqrt3) = x + x^2 + c`

• `4tan^-1((2y + 1)/sqrt3) = sqrt3/2(2x + x^2) + c`

• `sqrt3tan^-1((3y + 1)/sqrt3) = 4(1 + x + x^2) + c`

• `4tan^-1((2y + 1)/sqrt3) = sqrt3(2x + x^2) + c`

Answer 1

The solution of `dy/dx = 1 + y + y^2 + x + xy + xy^2` is `underline(4tan^-1((2y + 1)/sqrt3) = sqrt3(2x + x^2) + c)`.

Explanation:

`dy/dx = 1 + y + y^2 + x + xy + xy^2`

⇒ `dy/dx = (1 + y + y^2) (x + 1)`

Integrating on both sides, we get

`intdy/(1 + y + y^2) = int(x + 1)dx + c_1`

⇒ `intdy/((y + 1/2)^2 + 3/4) = x^2/2 + x + c_1`

⇒ `1/(sqrt3"/"2) . tan^-1((y + 1/2)/(sqrt3"/"2)) = x^2/2 + x + c_1`

⇒ `4tan^-1((2y + 1)/sqrt3) = sqrt3(x^2 + 2x) + c`,

where c = `2sqrt3  c_1`